Chop mushrooms in food processor (finely chopped) and transfer them to saucepan. Cover and cook over medium-high heat, stirring often, until juices are released. Bring liquid to a boil and reduce heat to simmer for 10 to 15 minutes.

Drain mushrooms in a strainer over a large bowl to catch strained liquid and let mushrooms cool slightly. You will have about 2 cups of mushroom puree and 5 cups of mushroom juice. Don’t worry if you have less, you will only need about 1 cup of puree.

Combine 1 cup of mushroom juice, 3/4 cup of heavy cream, and shallots in a medium saucepan. Mince garlic into the pan. Bring mixture to a simmer and cook, stirring occasionally, until liquid is reduced by about half to slightly more than 1 cup, about 5 minutes. Put 3/4 cup of mushroom puree in a small bowl. Stir 6 tbsp of reduced juice mixture into the puree. Season with salt and pepper.

Bring remaining mushroom juice to a simmer in a medium saucepan for 15 minutes. Add remaining 1/4 cup of heavy cream and soy sauce. Simmer for 5 more minutes and reduce heat to low to keep warm for serving.

Lay 4 slices of brioche and trim slices of cheese to fit the bread. Place 1 slice of cheese on each of the 4 slices of bread. Spread sandwich filling on top of cheese. Place another slice of cheese over filling and cover with another slice of bread.

Melt butter in a small saucepan and brush it on both sides of the sandwiches. Heat a large nonstick skillet over medium heat. When pan is hot, add the sandwiches and brown on both sides. Work in batches if necessary.

Pour warm dipping sauce into 4 small bowls. Serve sandwiches with dipping sauce on the side.